Special correspondent John Miller, who disclosed that he once worked for the National Intelligence, sat down with a NSA analyst who walked 60 Minutes through the process of "call chaining" — and allowed the cameras to look over his shoulder while doing so.

Pro-privacy demonstrators march in front of the US Capitol building on Oct. 26, 2013.

The NSA "Call Chaining" Method

The agency can "only target the communications of a US person with a probable cause under a specific court order," Alexander explained. According to the director, less than 60 people considered US persons around the world are being targeted. The NSA does not need a court order to target foreigners.

First, an analyst enters a phone number into the system. Only numbers that have been authorized for that particular analyst are accessible via his or her log-in credentials.

The analyst can then see that target's phone records from the last 18 days. This method is called "chaining out" a number.

Call history is represented in the form of a web. If another NSA target appears in the call history, then the analyst can "chain out" that target as well, and view the two webs side by side.

This information is helpful if, for example, two pirates are calling the same numbers. These numbers can provide information on safe houses or bigger targets up the chain.

The NSA adopted this method after the Sept. 11 attacks. Two of the involved hijackers were communicating with an Al-Qaeda safehouse in Yemen at the time, and the US intelligence agency was not able to detect that those calls from the hijackers were coming from California.

According to the segment, super computers that use "more power than most mid-sized cities" are required to sort through the metadata collected by the NSA. While only 50 to 60 US persons are targeted by the agency, data from over 300 million Americans is collected and analyzed by these systems. NSA agents do not listen in on calls or see the names that own those phone numbers, but in August, FISA, the secret court that presides over NSA cases, ruled that the agency overstepped its authorization many times.

Watch 60 Minutes's "Inside the NSA" episode below, then take our poll: is the NSA's "call chaining" method a breach of US citizens' privacy — or is it a necessary measure for protecting national security?